Woot. We did decide to get a rental car so maybe a quick stop before our arrival may be in order.
As for your kitty and doing it while the kids are away, might I suggest you have them say goodbye *just in case* he passes while they are gone. This way they can have some closure for the inevitable and wont be upset about the fact they didn't say goodbye, even though they will still be sad, at least its *something*.
My mom got rid of one of my favorite cats as a kid while I was away at my aunts and it's been well over 30 years and I am STILL resentful. Maybe it's me holding one heck of a grudge but if I knew I would have at least been able to part ways vs being frustrated all these years over it.

My BFF also had a similar experience with her elderly dog recently and put him down while the kids were in school. They fibbed to the kids about coming home and finding her and then bringing to the vet to try to help when in fact it was the opposite (planned to be put to sleep). Her kids are the type that ask 100's of questions so what she thought would be easiest has turned into one big lie that they keep adding onto trying to have answers that make sense to the kids. Not to mention playing cover up with the rest of the family that knows the truth of the matter.

Good luck with however you decide. It's not easy being a parent and having to deal with loss of your furry ones for you or the kids!